Numerical and experimental investigation on existing structures: two seminars

STRUCTURAL OPTIMISATION AND
INVERSE ANALYSIS STRATEGIES FOR MASONRY STRUCTURES
Dr Corrado Chisari
CSM Group – Department of Civil and Environmental Engineering
Imperial College London

Outline Optimisation Calibration of model parameters
Conclusions and ongoing
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Outline
 Optimisation
 Overview
 Genetic Algorithms
 Structural Optimisation
 Examples
 Calibration of model parameters
 Calibration problems as optimization problems
 Ill-posedness of inverse problems
 Identification of mesoscale model parameters for masonry
 Conclusions and ongoing research
Structural optimisation and inverse analysis strategies for masonry structures 2

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Overview
 Optimisation is the discipline that, starting from:
- the input variable space
- a model of the problem
 tries to find the best solution considering
- some objectives to achieve
- some constraints to satisfy.

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Overview
In mathematical terms
𝒙 = arg min
𝒙
𝜔1(𝒙), 𝜔2(𝒙), … , 𝜔𝑠(𝒙)
subjected to:
𝑔𝑗 𝒙 < 0, 𝑗 = 1, … , 𝑚
ℎ 𝑘 𝒙 = 0, 𝑘 = 1, … , 𝑛
where:
- 𝒙 input vector, 𝒙 ∈ 𝛀 (input variable space)
- 𝜔𝑖(𝒙) i-th objetive to minimise
- 𝑔𝑗(𝒙) j-th inequality constraint
- ℎ 𝑘 𝒙 k-th equality constraint

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Overview
When S=1 (mono-objective optimisation), some classical approaches are:
 Linear programming:
𝒙 = argmin
𝒙
𝒄 𝑻
𝒙 with 𝑨𝒙 ≤ 𝒃
 Quadratic programming:
𝒙 = argmin
𝒙
𝟏
𝟐
𝒙 𝑻
𝑸𝒙 + 𝒃𝒙 + 𝒄
 Constrained quadratic programming:
𝒙 = argmin
𝒙
𝟏
𝟐
𝒙 𝑻
𝑸𝒙 + 𝒃𝒙 + 𝒄 with 𝑨𝒙 ≤ 𝒒
 Convex programming:
𝒙 = argmin
𝒙
𝜔(𝒙) with 𝑔𝑖(𝒙) ≤ 0 and 𝜔(𝒙), 𝑔𝑖(𝒙) convex functions
Under some strict hypotheses, these types of problems have closed-formn solutions that can be obtained by means of well-
established methods (Simplex, Lagrange multipliers, ...).

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Overview
When the problem complexity increases iterative approaches:
𝒙 𝒕 = 𝒙 𝒕−𝟏 + Δ𝒙 𝒕 such that 𝜔 𝒙 𝒕 ≤ 𝜔 𝒙 𝒕−𝟏
They differ for the method to find the corrections Δ𝒙 𝒕:
 Line search with steepest descend (Jacobian);
 Line search with Newton direction (Hessian);
 Trust region.
They determine the path to follow by examining derivatives for ω (Jacobian and Hessian).
For this reason they are called gradient-based methods.

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Overview
Real-world problems does not usually satisfy one or more requirements for the mentioned
methods:
 The objective function is not convex
 The objective function and/or the constraints are not continuous
 The function and/or the constraints are not differentiable
 The variables x are discrete
 Multi-objective problem
 Black-box problem
Need for more general optimisation methods

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Genetic Algorithms
 They mimic the search for the optimum as observed in nature:
 A species evolves during a number of generations improving its fitness towards environmental conditions,
through recombination of genetic heritage of fittest individuals;
 The least fit individuals become extinct during the evolution;
 Casual mutations may bring novelties in an individual’s chromosome which, if positive, may propagate
and open new evolutions paths;
 It can happen that parents survive to their own offspring if these are not apt to the external environment
(elitism).
They belongs to the algorithm classes:
zero-order: they do not use derivatives
population-based: iterations regard populations, not just one
individual
stochastic: the process depends on some random components

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Genetic Algorithms
•Definition of chromosome and representation
•Definition of the fitness function
•Setting GA parameters
Creation of the first population
Stopping criterion satisfied?
Ranking of the population
Elitism
New population
Selection
Recombination (crossover)
Mutation
Optimum
Evaluation of the population
yes
no
x1 x2 x3 x4 x5 x6

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Multi-objective optimisation
𝒙 = arg min
𝒙
𝝎 con 𝝎 =
𝜔1(𝒙)
…
𝜔𝑠(𝒙)
 The classical approach involves scalarising vector 𝝎 by means of weights wi:
𝒙 = arg min
𝒙
𝜔 𝑠𝑐𝑎𝑙 with 𝜔 𝑠𝑐𝑎𝑙 = 𝒘 𝒕
𝝎
 The result depends on the choice of the weights.

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Multi-objective optimisation
 A solution x1 is said to dominate solution x2 (𝒙 𝟏 ≻ 𝒙 𝟐) if and only if:
 𝜔𝑖 𝒙 𝟏 ≤ 𝜔𝑖 𝒙 𝟐 ∀𝑖 = 1, … , S
 𝜔j 𝒙 𝟏 < 𝜔j 𝒙 𝟐 ∃𝑗 = 1, … , S
 The set of non-dominated solutions is called Pareto Front (PF).
 «Dominates» and «Pareto Front solution» are the
extensions of «is better than» and «optimal solution» to
the case of multiple objectives.
 Under some hypotheses, the solution of the scalarised
problem belongs to the PF. This is however the general
solution of the multi-objective problem.

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Non-dominated Sorting Genetic Algorithm (NSGA-II)
 Population-based problems are naturally suited to looking for an ensemble of
solutions (the Pareto Front) instead of a single solution
0
1000
2000
3000
4000
5000
6000
7000
0 500 1000
f2
f1
Initial population
0
200
400
600
800
1000
1200
1400
20 40 60 80
f2
f1
Converged population
Only the ranking method needs to be modified.
x1 ω(x1)
x2 ω(x2)≥ω(x1)
...
xN ω(xN)≥ω(xN-1)
x1 -
x2 𝒙 𝟏 ≽ 𝒙 𝟐
...
xN 𝒙 𝑵−𝟏 ≽ 𝒙 𝑵
Mono-objective Multi-objective

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Optimal design
Traditional design
Optimal design
Start Preliminary design
External action
analysis
Internal stresses
Section definition
Verification
Design
modification
End
NoYes
Start Parameterisation Trial structure
External action
analysis
Internal stresses
ObjectivesEnd
No
Yes
Verification
Optimisation
process

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Structural optimisation
Problem Variables Objectives Constraints Example
Parametric
optimisation
- Dimensions and
features of structural
elements
- Cost minimisation
- Optimising
performances
- Structural
constraints
- Operative
constraints
Topological
optimisation
- Parameters identifying
the shape of the
structure
- Weight
minimisation
- Stress
maximisation
- Structural
constraints
- Operative
constraints
Structural
identification
- Material parameters
- Unknown boundary
conditions
- Minimising
discrepancy with
experimental data
- Constraints due
the physical or
mathematical
nature of the
parameters
Input variables Control code
Output
variables

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Example 1 – Rastrigin function
Funzione di Rastrigin
𝑓 𝑥 = 𝐴𝑛 +
𝑖=1
𝑛
𝑥𝑖
2
− 𝐴𝑐𝑜𝑠(2𝜋𝑥𝑖
A=10,
𝑥𝑖 ∈ [−4.348,4.048]
𝑛 = 10 (numero di variabili)
𝒙 =
0.0115
0.0057
0.008
0.0101
−0.0012
−0.0024
0.0132
0.0087
−0.0055
0.0037
𝒙 𝑡𝑟𝑢𝑒 =
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

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Example 2 – Optimal design of bridges
Impalcati ottimi individuati
Spessori non vincolati Spessori vincolati
H=cost. H=var. H=cost. H=var.
Carpenteria [t] 760 769 840 733
Armatura L [t] 166 137 106 133
Calcestruzzo [t] 3168 3451 3168 3450
Costo [€] 2.562.089 2.551.675 2.802.303 2.492.916

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Example 3 – Optimal design of TMD for timber buildings
Copyright Maurer
Frequency
optimisation
Damping
optimisation
𝜇 =
𝑚 𝑇𝑀𝐷
𝑚 𝑠
𝛼 =
𝜔 𝑇𝑀𝐷
𝜔𝑠
𝜉2 =
𝑐 𝑇𝑀𝐷
2 𝑚 𝑇𝑀𝐷 𝜔 𝑇𝑀𝐷
This approach does
not account for the
external input

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Accounting for higher modes and more complex configurations

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Example 4 – Design of nonlinear viscous dampers
𝑭 = 𝒄 ∙ 𝒖 𝜶
∙ 𝒔𝒈𝒏( 𝒖)
S. Silvestri, G. Gasparini, T. Trombetti. A five-step procedure for the dimensioning of viscous
dampers to be inserted in building structures. J Earthq Eng, 14 (3) (2010), pp. 417-447

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Example 4 – Design of nonlinear viscous dampers
Advantages:
a. Flexibility in the objective
definition;
b. Possibility of embedding real-
world constraints;
c. Control on the forces transferred
to the structure.

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Example 5 – Optimal sensor layout for structural parameter
identification

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Example 6 - Performance-based design of FRP retrofitting of
existing RC frames
Braga, F., R. Gigliotti and M. Laterza, 2006. Analytical stress-
strain relationship for concrete confined by steel stirrups and/or
FRP jackets. J. Struct. Eng., 132: 1402-1416.
D’Amato, M., F. Braga, R. Gigliotti, S. Kunnath and M. Laterza, 2012.
A numerical general-purpose confinement model for non-linear
analysis of R/C members. Comput. Struct., 102-103: 64-75.
Strength increase
Ductility increase
Effect of confinement: triaxial state of stress
Toshimi Kabeyasawa, “Recent
Development of Seismic Retrofit
Methods in Japan”, Japan Building
Disaster Prevention Association, January,
2005.
FRP fabrics

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existing RC frames

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Inverse problems
 Given:
• The model space M;
• The data space D;
• The forward operator 𝒈(∙) ;
• Some observations 𝒅 𝒐𝒃𝒔 ∈ 𝑫;
Forward
operator
g (m)
Observable
data
d
Inverse
operator
g-1(d)
Model
parameters
m
Find 𝒎 ∈ 𝑴 such that:
𝒅 𝒐𝒃𝒔 = 𝒈( 𝒎)

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Solution of the inverse problem
 Since the analytical expression of 𝒈(∙) (and 𝒈−𝟏
(∙)) is generally not known, the problem
𝒎 = 𝒈−𝟏
(𝒅 𝒐𝒃𝒔) is replaced by
 𝒎 = arg min
𝒎∈𝑴
𝒅 𝒐𝒃𝒔 − 𝒈(𝒎) 𝑝
𝑝
Fernández-Martínez, J., Fernández-Muñiz, Z., Pallero, J. & Pedruelo-González, L., 2013. From Bayes to Tarantola: New insights to understand
uncertainty in inverse problems. Journal of Applied Geophysics, Volume 98, pp. 62-72.

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Identification problems
mtrial dc=g(mtrial) 𝜔 = 𝒅 𝒄 − 𝒅 𝒐𝒃𝒔
𝑡
𝑾 𝒅 𝒄 − 𝒅 𝒐𝒃𝒔
𝜔 𝑚𝑖𝑛?
Updating
m
Optimisation
algorithm
No
𝒎
Yes
dobs

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Identification of base-isolated bridges
FE model: modal analysis
Real structure: dynamic tests
𝜔2,𝑇 𝒑 =
𝑖=1
𝑁 𝑀
𝑓𝑟𝑒𝑓
∆𝑓𝑖
𝑇𝑖,𝑐𝑜𝑚𝑝(𝒑) − 𝑇𝑖,𝑒𝑥𝑝
𝑇𝑒𝑥𝑝,𝑚𝑎𝑥
2
𝜔2,𝑀𝐴𝐶 𝒑 =
𝑖=1
𝑁 𝑀
𝑇𝑖,𝑒𝑥𝑝
𝑇𝑒𝑥𝑝,𝑚𝑎𝑥
1 − max
𝑗
(𝑀𝐴𝐶𝑖𝑗(𝒑)) 2
Chiara Bedon, Antonino Morassi, Dynamic testing and parameter identification of a
base-isolated bridge, Engineering Structures, Volume 60, 2014, 85–99

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Identification of base-isolated bridges

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Ill-posedness of the inverse problem
Well-posed problem (Hadamard, 1902):
1. The solution exists;
2. It is unique;
3. It is stable.
Input Output
Input Output
Forward
problem
Inverse
problem
While the forward problem is usually well-
posed, it is not always the case of the
corresponding inverse problem.

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Calibration of a model for steel members
Monotonic test
Cyclic
test (AISC
protocol)
Pseudo-dynamic
test (Spitak ground
motion)
S1
S2
S3
S4
Numerical model: smooth model,
implemented in SeismoStruct
(M. V. Sivaselvan and A. M. Reinhorn,
“Hysteretic models for deteriorating
inelastic structures,” J. Eng. Mech., vol. 126,
no. 6, pp. 633-640, 2000)

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Multiple responses
Danger of overfittingModels are never perfect

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Mono- vs multi-objective optimisation
Calibration response Validation response
Calibration by mono-objective optimisation
Calibration by bi-objective optimisation

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Motivation
A. Borri, G. Castori, M. Corradi, E. Speranzini, Shear behavior
of unreinforced and reinforced masonry panels subjected to in
situ diagonal compression tests, Construction and Building
Materials, 25(12), 2011, 4403 – 4414.
Tests for macro-models are very invasive
M. Corradi , A. Borri , A. Vignoli, Experimental study on the
determination of strength of masonry walls, Construction and Building
Materials, 17(5), 2003, 325 – 337.

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Motivation
Tests on single components are
less invasive…
…but it is difficult to extract
representative specimens from
existing structures
http://www.matest.com/
≠

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Features of the test
• To be performed in-situ (existing structures)
• Low-invasive
• Involving a sufficient volume of masonry
• Able to capture elastic and strength material parameters for interfaces

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Flat-jacks
http://www.expin.it/servizi/indagini-strutturali/?lang=en

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Flat-jack test – preliminary study
The experimental setup consists of
different phases:
1. Horizontal cut;
2. Horizontal flat-jack;
3. Two vertical cuts and restraining;
4. Vertical flat-jack.

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kVkN

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• Instrumental layout
• Noise propagation
• Assessment of results

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• The effect of horizontal cut/flat jack
• Horizontal tension
• Yielding at the border
• Decreasing with distance
𝜎𝑥 = 𝑝
sinh 2𝜉 − 2
2𝑒 𝜉 cosh3 𝜉
𝜎 𝑦 = 𝑝 tanh3
𝜉
𝜎 𝑥 = 𝑝
sinh2
𝜉
𝑒 𝜉 cosh3 𝜉
𝜎 𝑦 = 𝑝 coth 𝜉
𝑢 𝑥 = −
3𝑁 𝑚
𝐸𝑡ℎ𝛽(1 + 2𝑘)
sinh 𝛽𝑥 +
3𝑁 𝑚
tanh 𝛽𝑏1 cosh 𝛽𝑥
𝑢 𝑥 = −
3𝑁 𝑚
sinh 𝛽𝑥 +
3𝑁 𝑚
coth 𝛽𝑏1 cosh 𝛽𝑥
• The effect of vertical flat jack
• Horizontal deformability
• Boundary conditions

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Flat-jack test – a new setup

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The flat-jack test - instruments

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The flat-jack test – test 1

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 Bricks in tension:
1. Micro-cracking leads to premature decreasing of stiffness that
is difficult to identify;
2. A vertical crack propagates without involving mortar joint
nonlinear behaviour.
Local reinforcement by means of CFRP strips

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Elastic parameters
Parameter Lower bound Upper bound Step Solution
𝑘 𝑉 10 N/mm3 4000 N/mm3 1 N/mm3
443 N/mm3
𝑟ℎ𝑗,𝑏𝑗 0.001 0.7 10-4
0.1
𝑘 𝑁 10 N/mm3 4000 N/mm3 1 N/mm3
218 N/mm3
kVkN
Solutions
𝑡𝑜𝑙 = 𝑘2
𝜔 𝑟𝑒𝑓
k=10%
𝜔 𝑟𝑒𝑓 = 𝒖 𝒐𝒃𝒔
𝑡
𝑾𝒖 𝒐𝒃𝒔
Eb=11200 MPa
υb=0.19
(Experimental)

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The flat-jack test - strength parameters
Interface Property Symbol
Simplified
assumption
Bedjoints
initial cohesion 𝑐0,𝑏𝑗 -
initial friction coefficient tan 𝜙0,𝑏𝑗 -
initial tensile strength 𝜎𝑡0,𝑏𝑗
𝑐0,𝑏𝑗
20
+ 0.25
𝑐0,𝑏𝑗
𝑡𝑎𝑛 𝜙0,𝑏𝑗
residual cohesion 𝑐 𝑟,𝑏𝑗 0
residual friction coefficient tan 𝜙 𝑟,𝑏𝑗 tan 𝜙0,𝑏𝑗
residual tensile strength 𝜎𝑡𝑟,𝑏𝑗 0
energy fracture, mode I 𝐺𝑓𝐼,𝑏𝑗 Very high
energy fracture, mode II 𝐺𝑓𝐼𝐼,𝑏𝑗 Very high
energy fracture, compression 𝐺𝑓𝐶,𝑏𝑗 Very high
dilatancy angle tan 𝜙 𝑑,𝑏𝑗 0
Headjoints
initial cohesion 𝑐0,ℎ𝑗 0
initial friction coefficient tan 𝜙0,ℎ𝑗 tan 𝜙0,𝑏𝑗
initial tensile strength 𝜎𝑡0,ℎ𝑗 0
residual cohesion 𝑐 𝑟,ℎ𝑗 0
residual friction coefficient tan 𝜙 𝑟,ℎ𝑗 tan 𝜙 𝑟,𝑏𝑗
residual tensile strength 𝜎𝑡𝑟,ℎ𝑗 0
energy fracture, mode I 𝐺𝑓𝐼,ℎ𝑗 Very high
energy fracture, mode II 𝐺𝑓𝐼𝐼,ℎ𝑗 Very high
energy fracture, compression 𝐺𝑓𝐶,ℎ𝑗 Very high
dilatancy angle tan 𝜙 𝑑,ℎ𝑗 0
Unknowns:
• 𝑐0,𝑏𝑗
• tan 𝜙0,𝑏𝑗=tan 𝜙 𝑟,𝑏𝑗

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Strength parameters – simplified approach
𝑝 𝑢 ⋅ ℎ = 𝜏 𝑢 ⋅ 𝑏
tan 𝜙 𝑟 =
𝜏 𝑢
𝜎𝑣
=
𝑝 𝑢
𝜎𝑣
ℎ
𝐵
≅ 1.0

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Strength parameters – simplified approach
rhj,bj
kV
(N/mm3)
kN
(N/mm3)
c0,bj
(MPa)
ω
0.0 1061 333 0.454 0.001014
0.1 669 221 0.559 0.001201
0.2 578 185 0.606 0.0014
0.3 465 171 0.641 0.001628
0.4 348 174 0.670 0.001889
0.5 275 179 0.696 0.002233
0.6 256 170 0.720 0.002506
Optimum
𝑘 𝑉 = 1061𝑁/𝑚𝑚3
𝑘 𝑁 = 333𝑁/𝑚𝑚3
𝑟ℎ𝑗,𝑏𝑗 = 0.0
𝑐0,𝑏𝑗 = 0.454
tan 𝜙0,𝑏𝑗 = 1.0

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Validation – phase 1

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Validation – test 1
(a) (b)
first
crack
𝒑 𝒗𝒇𝒋 = 𝟏. 𝟏𝑴𝑷𝒂
𝒑 𝒗𝒇𝒋 = 𝟏. 𝟏𝟔𝑴𝑷𝒂𝒑 𝒗𝒇𝒋 = 𝟎. 𝟗𝟗𝑴𝑷𝒂

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URM perforated wall
𝑟ℎ𝑗,𝑏𝑗 = 0.6 (𝛿ℎ = 1.3 𝑚𝑚) 𝑟ℎ𝑗,𝑏𝑗 = 0.0 (𝛿ℎ = 1.3 𝑚𝑚) 𝑟ℎ𝑗,𝑏𝑗 = 0.0 (𝛿ℎ = 5 𝑚𝑚)
Elastic Stiffnesses
(kN/mm)
rhj,bj=0.0 231.08 10%
rhj,bj=0.1 209.20 0%
rhj,bj=0.2 197.70 -6%
rhj,bj=0.3 191.50 -9%
rhj,bj=0.4 191.88 -9%
rhj,bj=0.5 192.26 -8%
rhj,bj=0.6 188.54 -10%
Central
value 209.81
rhj,bj
kV
(N/mm3)
kN
(N/mm3)
c0,bj
(MPa)
0.0 1061 333 0.454
0.1 669 221 0.559
0.2 578 185 0.606
0.3 465 171 0.641
0.4 348 174 0.670
0.5 275 179 0.696
0.6 256 170 0.720

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Conclusions and ongoing research
 Full characterisation by means of flat-jacks
 Meta-model approximation for expensive simulations
 Use of full-field measurements
 Multi-level model calibration (MultiCAMS project)

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References
 Chisari C, Bedon C, 2017. Performance-based design of FRP retrofitting of existing RC frames by means of multi-objective optimisation. Bollettino
di Geofisica Teorica e Applicata, 58(4), pp. 377-394.
 Parcianello E, Chisari C, Amadio C, 2017. Optimal design of nonlinear viscous dampers for frame structures. Soil Dynamics and Earthquake
Engineering 100:257-260.
 Chisari C, Macorini L, Amadio C, Izzuddin BA, 2016. Optimal sensor placement for structural parameter identification. Structural and
Multidisciplinary Optimization, 55(2): 647-662, DOI: 10.1007/s00158-016-1531-1.
 Chisari C, Francavilla AB, Latour M, Piluso V, Rizzano G, Amadio C, 2017. Critical issues in parameter calibration of cyclic models for steel members.
Engineering Structures, 132:123-138, DOI: 10.1016/j.engstruct.2016.11.030.
 Chisari C, Bedon C, 2016. Multi-objective optimization of FRP jackets for improving seismic response of reinforced concrete frames. American
Journal of Engineering and Applied Sciences 9(3): 669-679, DOI:10.3844/ajeassp.2016.669.679.
 Poh’sié GH, Chisari C, Rinaldin G, Amadio C, Fragiacomo M, 2016. Optimal design of tuned mass dampers for a multi-storey cross laminated
timber building against seismic loads. Earthquake Engineering and Structural Dynamics, DOI: 10.1002/eqe.2736.
 Chisari C, Macorini L, Amadio C, Izzuddin BA, 2015. An Experimental-Numerical Procedure for the Identification of Mesoscale Material Properties
for Brick-Masonry. Proceedings of the 15th International Conference on Civil, Structural and Environmental Engineering Computing, Paper 72
 Chisari C, Bedon C, Amadio C, 2015. Dynamic and static identification of base-isolated bridges using Genetic Algorithms. Engineering Structures
102:80-92.
 Poh'sié GH, Chisari C, Rinaldin G, Fragiacomo M, Amadio C, Ceccotti A, 2015. Application of a translational tuned mass damper designed by means
of genetic algorithms on a multistory cross-laminated timber building. Journal of Structural Engineering ASCE, DOI: 10.1061/(ASCE)ST.1943-
541X.0001342, E4015008.
 Chisari C, Macorini L, Amadio C, Izzuddin BA, 2015. An Inverse Analysis Procedure for Material Parameter Identification of Mortar Joints in
Unreinforced Masonry, Computer and Structures 155:97-105.
 Amadio C, Chisari C, Panighel F, 2015. Analysis and optimisation of tapered steel-concrete composite bridges. Proceedings of the
25th Conference on Steel Structures C.T.A., Salerno (in Italian).

THANK YOU FOR YOUR
ATTENTION
corrado.chisari@gmail.com
c.chisari12@imperial.ac.uk

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